| Let Ω be a bounded open domain in Rn with smooth boundary(?)Ω,X ＝(X1,X2,...,Xm)be a system of real smooth vector fields defined on Ω and the boundary(?)Ω is non-characteristic for X.If X satisfies the Hormander’s condition,then the vector fields is finitely degenerate and the sum of square operator △x =∑j=1m Xj2 is a finitely de-generate elliptic operator.If λj is the jth Dirichlet eigenvalue for-△x on Ω,then this paper shall study the lower bound estimates for λj.In this paper,we shall study the precise lower bound estimate of the Dirichlet eigenvalue for some kinds of general Grushin type degenerate elliptic operators △x on Ω First,we study a Grushin type vector field X=((?)x1,…,(?)xn-1,f(x)(?)xn),x=(x1,...,xn-1)= where f(x)represents some polynomial without constant term.And f(x)has a unique zero point at origin x ＝ 0 in Ω.In this paper,we give the lower bound estimate of the Dirichlet eigenvalues for the sum of square operator △X,whereand BQ = min{1,n(3-Q)/2 },C is the constant appeared in Proposition 2.0.1 below,Wn-1 iS the area of the unit sphere in Rn,|Ω|n is the volume of Ω.Then,we study a more general Grushin type vector field X=((?)x1,...,(?)xn-p,f1(x(p))(?)xn-p+1,...,fp(x(p))(?)xn)x(p)=(x1,...,xn-p),n-p<j<n,where fj has a unique zero point at origin xp=0 in Ω.In this paper,we give the lower bound estimate of the Dirichlet eigenvalues for the sum of square operator △x,whereC is the constant appeared in Proposition 2.0.1 below,ωn-1 is the area of the unit sphere in Rn,|Ω|n is the volume of Ω. |
